Significant Figures Multiplication And Division Practice

Significant Figures: Multiplication and Division Practice – Mastering the Calculations

Significant figures (sig figs) are a crucial aspect of scientific notation and calculations. They dictate the precision of a measurement and directly impact the accuracy of results obtained through mathematical operations. While addition and subtraction rules for significant figures differ slightly, multiplication and division share a simpler, yet equally vital, set of guidelines. This comprehensive guide will provide you with a thorough understanding of significant figures in multiplication and division, complete with practice problems and detailed explanations to solidify your mastery.

Understanding Significant Figures

Before delving into the intricacies of multiplication and division with sig figs, let's refresh our understanding of what constitutes a significant figure. Significant figures are the digits in a number that carry meaning contributing to its precision. Here's a quick recap of the rules:

• Non-zero digits are always significant. For example, in the number 25.8, all three digits are significant.

• Zeros between non-zero digits are always significant. In 1005, all four digits are significant.

• Leading zeros (zeros to the left of the first non-zero digit) are never significant. 0.0023 has only two significant figures (2 and 3).

• Trailing zeros (zeros to the right of the last non-zero digit) are significant only if the number contains a decimal point. 1200 has two significant figures, while 1200.0 has five.

• Trailing zeros in a number without a decimal point are ambiguous. Scientific notation is preferred to avoid this ambiguity.

Significant Figures in Multiplication and Division: The Rule

The rule for significant figures in multiplication and division is elegantly straightforward: The result of a multiplication or division calculation should have the same number of significant figures as the measurement with the fewest significant figures.

Let's illustrate this with an example:

Suppose you are calculating the area of a rectangle with length 12.5 cm and width 4.3 cm.

Area = Length × Width = 12.5 cm × 4.3 cm = 53.75 cm²

The length (12.5 cm) has three significant figures, while the width (4.3 cm) has only two. Following the rule, the area should be reported with only two significant figures. Therefore, the correct answer is 54 cm². We round up because the digit following the last significant figure (7) is greater than or equal to 5.

Practice Problems: Multiplication

Let's work through several practice problems to solidify your understanding of significant figures in multiplication. Remember, the key is to identify the number with the fewest significant figures and round your answer accordingly.

Problem 1:

Calculate the volume of a cube with sides of length 5.2 cm.

• Solution: Volume = 5.2 cm × 5.2 cm × 5.2 cm = 140.608 cm³

• Significant Figures: 5.2 cm has two significant figures. Therefore, the volume should be reported as 140 cm³ (rounded to two significant figures).

Problem 2:

A car travels at an average speed of 65.3 mph for 3.2 hours. What is the total distance traveled?

• Solution: Distance = 65.3 mph × 3.2 hours = 208.96 miles

• Significant Figures: 3.2 hours has two significant figures. The distance traveled should be reported as 210 miles (rounded to two significant figures).

Problem 3:

Calculate the area of a circle with a radius of 2.55 m. (Use π ≈ 3.14159)

• Solution: Area = π × (2.55 m)² = 20.428 m²

• Significant Figures: The radius has three significant figures. However, π (pi) has effectively infinite significant figures in this context. Therefore, the answer should have three significant figures. Thus, the area is reported as 20.4 m².

Problem 4:

A rectangular prism has dimensions of 10.0 cm, 2.5 cm, and 0.010 cm. Find its volume.

• Solution: Volume = 10.0 cm × 2.5 cm × 0.010 cm = 0.25 cm³

• Significant Figures: 2.5 cm and 0.010 cm have two and two significant figures respectively. Therefore, the final answer also has two significant figures. The volume is reported as 0.25 cm³.

Practice Problems: Division

Now, let's move on to practicing with division. The rules remain the same: count the number of significant figures in each value and round your final answer to the number of significant figures in the measurement with the fewest significant figures.

Problem 1:

A car travels 250 miles in 5.2 hours. What is its average speed?

• Solution: Speed = 250 miles / 5.2 hours ≈ 48.077 mph

• Significant Figures: 250 miles has two significant figures (the trailing zeros are ambiguous without a decimal), and 5.2 hours has two. Therefore, the average speed should be rounded to 48 mph.

Problem 2:

The mass of an object is 10.0 g and its volume is 2.5 cm³. What is its density? (Density = Mass/Volume)

• Solution: Density = 10.0 g / 2.5 cm³ = 4.0 g/cm³

• Significant Figures: Both values have two significant figures, resulting in a final answer of 4.0 g/cm³.

Problem 3:

A student measures the length of a wire to be 12.75 cm and its mass to be 0.0052 kg. Calculate the linear density (mass/length).

• Solution: Linear density = 0.0052 kg / 12.75 cm = 0.00040787 kg/cm

• Significant Figures: 0.0052 kg has two significant figures, and 12.75 cm has four. Thus the result must have two significant figures. Therefore, the linear density is 0.00041 kg/cm (or 4.1 x 10⁻⁴ kg/cm in scientific notation).

Problem 4:

The area of a circle is 78.54 cm². The radius can be calculated from this area using the formula: Area = πr². Calculate the radius of the circle. (Use π ≈ 3.14159)

• Solution: Radius = √(Area/π) = √(78.54 cm²/3.14159) ≈ 4.9996 cm

• Significant Figures: The area has four significant figures. Therefore, the radius should also have four significant figures. The result is 5.000 cm.

Advanced Considerations and Common Mistakes

1. Chain Calculations: When performing a series of multiplications and divisions, apply the significant figure rule at the very end of the calculation. Avoid rounding intermediate results, as this can accumulate errors.

2. Exact Numbers: Numbers that are defined or counted, not measured, are considered to have an infinite number of significant figures. For instance, the number 2 in the formula for the area of a circle (A = πr²) is an exact number and does not limit the significant figures in the calculation. Similarly, the number of sides in a triangle is an exact number.

3. Scientific Notation: Using scientific notation simplifies the handling of significant figures, particularly with very large or very small numbers. It removes ambiguity related to trailing zeros.

Conclusion

Mastering significant figures is essential for performing accurate scientific calculations. The rules for multiplication and division are relatively simple: the final answer should have the same number of significant figures as the measurement with the fewest significant figures. Through consistent practice and careful attention to detail, you can confidently handle significant figures in your calculations and ensure the precision and accuracy of your scientific work. Remember to always double-check your work and understand the underlying principles to avoid common mistakes and confidently interpret your results. Continuous practice, especially with varied problem types, is key to developing a robust understanding of this crucial concept.